The universal minimal system for the group of homeomorphisms of the Cantor set

E. Glasner, and B. Weiss. "The universal minimal system for the group of homeomorphisms of the Cantor set." Fundamenta Mathematicae 176.3 (2003): 277-289. <http://eudml.org/doc/282616>.

@article{E2003,
abstract = {Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E) of self-homeomorphisms of the Cantor set E, with the topology of uniform convergence, the universal minimal system (M(G),G) is isomorphic to Uspenskij’s “maximal chains” dynamical system (Φ,G) in $2^\{2^E\}$. In particular it follows that M(G) is homeomorphic to the Cantor set. Our main tool is the “dual Ramsey theorem”, a corollary of Graham and Rothschild’s Ramsey’s theorem for n-parameter sets. This theorem is used to show that every minimal symbolic G-system is a factor of (Φ,G), and then a general procedure for analyzing G-actions of zero-dimensional topological groups is applied to show that (M(G),G) is isomorphic to (Φ,G).},
author = {E. Glasner, B. Weiss},
journal = {Fundamenta Mathematicae},
keywords = {minimal dynamical system; homeomorphisms; Cantor set; dual Ramsey theorem},
language = {eng},
number = {3},
pages = {277-289},
title = {The universal minimal system for the group of homeomorphisms of the Cantor set},
url = {http://eudml.org/doc/282616},
volume = {176},
year = {2003},
}

TY - JOUR
AU - E. Glasner
AU - B. Weiss
TI - The universal minimal system for the group of homeomorphisms of the Cantor set
JO - Fundamenta Mathematicae
PY - 2003
VL - 176
IS - 3
SP - 277
EP - 289
AB - Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E) of self-homeomorphisms of the Cantor set E, with the topology of uniform convergence, the universal minimal system (M(G),G) is isomorphic to Uspenskij’s “maximal chains” dynamical system (Φ,G) in $2^{2^E}$. In particular it follows that M(G) is homeomorphic to the Cantor set. Our main tool is the “dual Ramsey theorem”, a corollary of Graham and Rothschild’s Ramsey’s theorem for n-parameter sets. This theorem is used to show that every minimal symbolic G-system is a factor of (Φ,G), and then a general procedure for analyzing G-actions of zero-dimensional topological groups is applied to show that (M(G),G) is isomorphic to (Φ,G).
LA - eng
KW - minimal dynamical system; homeomorphisms; Cantor set; dual Ramsey theorem
UR - http://eudml.org/doc/282616
ER -